Yet another variant.
The disadvantage of the previous variants is the rigid assignment of the u values of the frame to the respective construction. Therefore, both variants are different regarding the position of the texture to the quadrants. Also, the texture is not applied symmetrically.
With 4 more triangles compared to the previous variant and the duplication of the positions in the middle of the four sides I solved the problem. Thereby I like even the quadrant-wise construction very much. It is clearer than the previous ones.

The texture can be wrapped neatly around the object.

Other variant

An additional optional parameter allows to select the starting quadrant for the texture.
RoundedBoxFlat( width, height, depth, radiusCorner, smoothness, uStartQuadr ); // uStartQuadr optional
function RoundedBoxFlat( w, h, d, r, s, q ) {
let qu = q || 1; // qu: start quadrant regarding u, optional
const pi = Math.PI;
let indices = [];
let positions = [];
let uvs = [];
makeFronts( s, 1, 0 ); // smoothness, front is 1, start index center front
makeFronts( s, -1, 4 * ( s + 3 ) + 1 ); // smoothness, back is -1, start index center back
makeFrame( s, 2 * ( 4 * ( s + 3 ) + 1 ), 1, 4 * ( s + 3 ) + 2 ); // smoothness, start index framing ,start index front, start index back
const geometry = new THREE.BufferGeometry( );
geometry.setIndex( new THREE.BufferAttribute( new Uint32Array( indices ), 1 ) );
geometry.setAttribute( 'position', new THREE.BufferAttribute( new Float32Array( positions ), 3 ) );
geometry.setAttribute( 'uv', new THREE.BufferAttribute( new Float32Array( uvs ), 2 ) );
// add multimaterial groups for front, back, framing
const vtc = 4 * ( s + 2 ) * 3;
geometry.addGroup ( 0, vtc, 0 );
geometry.addGroup ( vtc , vtc, 1 );
geometry.addGroup ( 2 * vtc, 2 * vtc + 3, 2 );
geometry.computeVertexNormals( );
return geometry;
function makeFronts( s, side, idx ) {
const d0 = side === 1 ? 0 : 1;
const d1 = side === 1 ? 1 : 0;
let id = 0;
for ( let q = 1; q < 5; q ++ ) { // quadrants
id ++;
for ( let j = 0; j < s + 2; j ++ ) {
indices.push( idx, idx + d0 + id, idx + d1 + id );
id ++;
}
}
positions.push( 0, 0, side * d / 2 ); // center
uvs.push( 0.5, 0.5 );
let x, y, z, sgnX, sgnY;
let phi = 0;
const u0 = side === 1 ? 0 : 1;
for ( let q = 1; q < 5; q ++ ) {
sgnX = q === 1 || q === 4 ? 1 : -1;
sgnY = q < 3 ? 1 : -1 ;
x = Math.cos( phi ) * w / 2;
y = Math.sin( phi ) * h / 2;
z = side * d / 2;
positions.push( x, y, z );
uvs.push( u0 + side * ( 0.5 + x / w ), 0.5 + y / h );
for ( let j = 0; j < s + 1; j ++ ) {
const c = { x: sgnX * ( w / 2 - r ), y: sgnY * ( h / 2 - r ), z: side * d / 2 } // quadrant center
const dPhi = pi / 2 * j / s;
x = c.x + r * Math.cos( phi + dPhi );
y = c.y + r * Math.sin( phi + dPhi );
z = c.z;
positions.push( x, y, z );
uvs.push( u0 + side * ( 0.5 + x / w ), 0.5 + y / h );
}
phi = phi + pi / 2;
x = Math.cos( phi ) * w / 2;
y = Math.sin( phi ) * h / 2;
z = side * d / 2;
positions.push( x, y, z );
uvs.push( u0 + side * ( 0.5 + x / w ), 0.5 + y / h );
}
}
function makeFrame( s, sidx, sif, sib ) {
let a, b, c, d, xf, yf, zf, xb, yb, zb;
const pif = sif * 3; // position start index front
const pib = sib * 3; // position start index back
let idx = sidx;
for ( let q = 1; q < 5; q ++ ) {
for ( let j = 0; j < s + 2; j ++ ) {
a = idx;
b = idx + 1;
c = idx + 2;
d = idx + 3;
indices.push( a, b, d, a, d, c );
idx += 2;
}
idx += 2;
}
const ls = 2 * r * Math.sin( pi / ( s * 4 ) ); // length of the outer line of a corner segment
const w2r = w / 2 - r;
const h2r = h / 2 - r;
const peri = 4 * w2r + 4 * h2r + 4 * s * ls; // perimeter
let u;
idx = 0; // reset
for ( let q = 1; q < 5; q ++ ) {
// console.log ( 'qu', qu );
u = qu / 4;
for ( let j = 0; j < s + 3; j ++ ) {
xf = positions[ pif + idx ];
yf = positions[ pif + idx + 1 ];
zf = positions[ pif + idx + 2 ];
xb = positions[ pib + idx ];
yb = positions[ pib + idx + 1 ];
zb = positions[ pib + idx + 2 ];
positions.push( xf, yf, zf, xb, yb, zb );
idx += 3;
// console.log ( 'u ', u );
uvs.push( u , 0, u, 1 );
if ( j === 0 ) { u -= q === 1 || q === 3 ? h2r / peri : w2r / peri; }
if ( j === s + 1 ) { u -= q === 1 || q === 3 ? w2r / peri : h2r / peri; }
if ( j > 0 && j < s + 1 ) { u -= ls / peri; }
}
qu = 4 - ( ( 5 - qu ) % 4 ); // cyclic next quadrant with respect to u
}
}
}