Equality of weak and ordinary Boolean Ramsey numbers for balanced V-shaped posets

Let $\boldsymbol{P}_i=\mathbf{V}_{m_i,m_i}$ for $1\leq i\leq k$, where $\mathbf{V}_{m_i,m_i}$ denotes the balanced V-shaped poset with $m_i$ elements on each branch. Write $R_w(\boldsymbol{P}_1,\boldsymbol{P}_2,\ldots,\boldsymbol{P}_k)$ for the weak Boolean Ramsey number and $R(\boldsymbol{P}_1,\boldsymbol{P}_2,\ldots,\boldsymbol{P}_k)$ for the ordinary Boolean Ramsey number. The equality conjecture.

$$R_w(\boldsymbol{P}_1,\boldsymbol{P}_2,\ldots,\boldsymbol{P}_k)=R(\boldsymbol{P}_1,\boldsymbol{P}_2,\ldots,\boldsymbol{P}_k)=m_1+m_2+\cdots+m_k+1.$$

This extends the equality already observed for the examples arising from the lower-bound colorings and is proposed as a generalization of the preceding theorem to these balanced V-shaped posets. The claim is presented without a proof in the source.