Piecewise linear height-product caps can be replaced by one extremum

Use the supplied disk parameter to replace $D$ by a convex polygon $P$. Its product parameter and $J$ are finite PL after compatible prism subdivisions by [F2]. Choose $p\in \mathrm{Int}P$ and $0<c<1$, and let $K$ be the convex pyramid with apex $(p,c)$ and base $P\times \{1\}$ in $P\times [0,1]$. Its lateral boundary $G_0=(p,c)\ast (\mathrm{\partial }P\times \{1\})$ is a parametrized PL disk. It intersects the disk $F_0=(P\times \{0\})\cup (\mathrm{\partial }P\times [0,1])$ exactly in their common boundary $\mathrm{\partial }P\times \{1\}$. The latter is also a parametrized PL disk: it is the complement of the top face in the boundary of a convex prism, and mapping its bottom and side strips to concentric planar polygonal regions gives the finite disk parameter.

The pyramid meets the boundary of the prism exactly in its top base. The boundary-window charts along its base perimeter are ordinary linear dihedral-sector charts, including at polygon corners after a finite angular subdivision; these identify the outer prism with a half-space and the inner pyramid with a quadrant. Thus [F3] gives a finite PL ball parameter on $W_0=\overline{(P\times [0,1])\setminus K}$. Its boundary is exactly $F_0\cup G_0$. Transfer these disks, the pyramid and $W_0$ to $Z$ by the product parameter and $J$. This gives the required $G$ and ball $W$ with boundary $F\cup G$.

Each horizontal section of $G_0$ above height $c$ and below height one is the boundary of a homothetic polygon $p+\frac{t-c}{1-c}(P-p)$. Thus its only interior height extremum is its apex minimum. At every other interior point its incident cone faces have nonzero height slope along their common generator, so a finite face-strip chart gives a local PL product with height as interval coordinate. Equivalently these strips are truncated convex cone faces, mapped to rectangular strips by matching their end edges and coning boundary maps with cone points at equal height. Composing with the height-preserving $J$ preserves these charts and the unique extremum. No conclusion about the height on the exterior disk $E$ is needed.

The disks $E,F,G$ have the same boundary and pairwise disjoint interiors, because $E\cap Z=C$ and step 1.1 gives the other intersection. Their parameters are supplied above. If $E\cup G$ bounds a supplied PL ball, then this ball and $W$ satisfy [F4], which supplies the ball bounded by $E\cup F$. Replacing $h$ by $-h$ gives the reversed-height statement. Every selection, cone, prism subdivision and window construction is finite, so no choice assumption is required.