Plumbing open books realizes contact connected sum

Let $(\Sigma_1,\psi_1)$ and $(\Sigma_2,\psi_2)$ be contact open books with properly embedded Lagrangian balls $L_1\subset\Sigma_1$ and $L_2\subset\Sigma_2$, each having Legendrian boundary. Write $P(\Sigma_1,\Sigma_2;L_1,L_2)$ for their plumbing, and let $(M_i,\alpha_i)$ be the contact manifolds supported by the two open books.

Plumbing conjecture. There are deformations $\tilde\psi_i$ of $\psi_i$, isotopic to $\psi_i$ as symplectomorphisms, such that the open book

$$\bigl(P(\Sigma_1,\Sigma_2;L_1,L_2),\tilde\psi_2\circ\tilde\psi_1\bigr)$$

supports the contact structure on

$$(M_1,\alpha_1)\#(M_2,\alpha_2).$$

In dimension three, the corresponding statement is a theorem of Torisu for the $2$-Murasugi sum. The higher-dimensional formulation is presented as conjectural and is intended to imply invariance under right-handed stabilization.