Burr–Erdős–Faudree–Rousseau–Schelp conjecture for star forests

Let $s$ and $t$ be positive integers. Let $n_{1}\geq n_{2}\geq \cdots\geq n_{s}\geq 1$ and $m_{1}\geq m_{2}\geq \cdots\geq m_{t}\geq 1$ be integers, and for each $k\in [s+t]\backslash [1]$ define

$$\ell_{k}=\max\left\{n_{i}+m_{j}-1:i+j=k\right\}.$$

Burr–Erdős–Faudree–Rousseau–Schelp conjecture. The size Ramsey number of the two star forests satisfies

$$\hat{r}\left(\bigsqcup_{i=1}^{s}K_{1,n_{i}},\bigsqcup_{j=1}^{t}K_{1,m_{j}}\right)=\sum_{k=2}^{s+t}\ell_{k}.$$

This conjecture was confirmed for uniform star forests and for several further cases, including when all $n_i$ and $m_j$ are odd or when $s=1$, but it remains open in general.