Convergence conjecture for deformations of several smoothable edges

Let $\mathsf{X}=\mathbb{P}^{n-1}\times\mathbb{D}^m$ and let $\pi(\varepsilon)=\sum_{k=0}^{\infty}\varepsilon^k\pi_k$ be the formal Poisson deformation whose existence is asserted for a semi-toric log symplectic form and a collection of smoothable edges, with each $\pi_k$ a holomorphic bivector on $\mathsf{X}$. Convergence conjecture. The power series

$$\pi(\varepsilon)=\sum_{k=0}^{\infty}\varepsilon^k\pi_k$$

can be chosen to be convergent, at least after shrinking the polydisc. This would upgrade the formal joint unobstructedness result for smoothable-edge Poisson deformations to an analytic deformation statement; the source does not provide a proof.