The decomposed-spider conjecture for graph burning

Let $T$ be a tree containing a decomposed spider formed by paths $\\{Q_i\\}_{i=1}^t$, where $t\geq 3$, and let $v_i$ be the non-terminal endpoint of $Q_i$ for $1\leq i\leq t$. Write $b(G)$ for the burning number of a graph $G$.

Decomposed-spider conjecture. If

$$b\left(\bigcup_{i=1}^t Q_i\right)\geq k$$

and

$$d(v_i,v_j)\geq 2k$$

for all $1\leq i,j\leq t$, then $b(T)\geq k+1$.

This conjecture proposes that a sufficiently separated decomposed spider forces the ambient tree to require at least one more burning step than the spider itself. The supplied text gives no resolution evidence, so its status is left open.