The conjecture on arbitrarily large Weinstein $\mathcal{L}$-invariants

The Weinstein domains in Examples 1 and 2 admit supporting multisections with divides, and their Weinstein $\mathcal{L}$-invariants are minimized over all such supporting multisections.

Arbitrarily large Weinstein $\mathcal{L}$-invariant conjecture. The Weinstein domains in Example 1 and Example 2 have arbitrarily large Weinstein $\mathcal{L}$-invariants under this minimization.

The preceding examples show that the invariant is nonzero when the relevant parameter satisfies $n>1$, but the claim that these Weinstein domains have arbitrarily large invariant remains open.