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

Let $p_1,p_2,\ldots,p_q$ be positive integers, and for each $i\in\{1,\ldots,q\}$ let

$$n_{i_1}\geq n_{i_2}\geq \cdots\geq n_{i_{p_i}}$$

be positive integers. Set $p=\sum_{i=1}^{q}p_i$, and let $F_1,F_2,\ldots,F_q$ be star forests with

$$F_i=\bigsqcup_{j=1}^{p_i}K_{1,n_{i_j}}.$$

For $k=q,\ldots,p$, define

$$l_k=\max\{(n_{1_{j_1}}-1)+(n_{2_{j_2}}-1)+\cdots+(n_{q_{j_q}}-1)+1:j_1+j_2+\cdots+j_q=k\}.$$

Multicolor star-forest conjecture. The multicolor size Ramsey number satisfies

$$\hat{r}(F_1,F_2,\ldots,F_q)=\sum_{k=q}^{p}l_k.$$

The equality is proposed as an extension of the two-color conjecture. The preceding construction establishes the corresponding upper bound, but the source gives no resolution of equality in general, so the conjecture is recorded as open.