Greedy optimality conjecture for tooth-dominant comb graphs

Let $C_{n,m}$ be the comb graph with spine parameter $n$ and tooth parameter $m$, let $b(C_{n,m})$ be its burning number, and let $T_{\mathrm{greedy}}^{(S)}$ be the number of fires used by the greedy algorithm when its initial placement is specified by $S$. Define

$$T_{\mathrm{greedy}}=\min\left\{T_{\mathrm{greedy}}^{(S)}:S=1,2,\ldots,n\right\}.$$

Greedy optimality conjecture for comb graphs. If $n\leq m$, then

$$b(C_{n,m})=\min\left\{T_{\mathrm{greedy}}^{(S)}:S=1,2,\ldots,n\right\}.$$

The greedy algorithm is proved optimal in the spine-dominant regime $n\geq m$, while in the tooth-dominant regime it is known to give an approximation and has been confirmed through $m=n+4$. Its optimality for all $n\leq m$ remains open.