The spacing-five conjecture for critical prism-canvases

A prism-canvas is a cylindrical fragment $(F,T_1,T_2,L)$ whose boundary faces are bounded by triangles $T_1,T_2$, with $|L(v)|\geq 5$ for every vertex outside the boundary triangles; it is critical when $F$ is $(T_1\cup T_2)$-critical with respect to $L$. Its spacing is the distance between the boundary triangles in the cylindrical fragment.

The spacing-five conjecture. There is no critical prism-canvas of spacing at least five.

A theorem cited in the paper gives an absolute upper bound on the spacing of critical prism-canvases, but the source notes that the explicit bound is much larger and presents spacing at least five as a conjectural strengthening.