Vanishing of the cotangent cohomology for flag combinatorial spheres

Let $\mathcal{K}$ be a combinatorial sphere and a flag complex. For every codimension-$2$ face $f$, let $\operatorname{lk}(f,\mathcal{K})$ denote its link, and let $A_{\mathcal{K}}$ be the Stanley–Reisner ring of $\mathcal{K}$.

Vanishing conjecture. If $\operatorname{lk}(f,\mathcal{K})$ is a $4$-gon or a $5$-gon for every codimension-$2$ face $f$, then

$$T^2_{A_{\mathcal{K}}}=0.$$

The conjecture is motivated by computations for flag $3$-spheres obtained by successive edge starrings, as well as by the corresponding result in dimension $2$. The stated computations found trivial $T^2$ in all $74$ examples considered, but the general assertion remains open.