Width conjecture for triangle-polygon graphs

Let $P_{t,k}=P_3|P_k$ be the graph with $k+1$ vertices obtained by gluing a triangle and a $k$-gon along an edge. Let $w^1_{{\mathcal A}_3}(G)$ be the width of the torsion in $H^{1,*}_{{\mathcal A}_3}(G)$, namely the maximal torsion grading minus the minimal torsion grading, with value $-1$ when all cohomology is free. Width conjecture for $P_{t,k}$. For these graphs,

$$w^1_{{\mathcal A}_3}(P_{t,k})=k-3.$$

The formula is based on the displayed computations for this family; the source does not state a resolution.