Conjugacy conjecture for Cartan subalgebras of tensor products

Let $V$ be a strongly rational vertex operator algebra, let $L$ be an even lattice, and let $V_L$ be the conformal vertex algebra associated with $L$. A Cartan subalgebra is understood in the sense used for conformal vertex algebras in the paper.

Cartan-subalgebra conjugacy conjecture. All Cartan subalgebras of the conformal vertex algebra $V\otimes V_L$ are conjugate.

If true, this would show that the more restrictive hyperbolic-genus definition requiring an isomorphism to preserve chosen Cartan subalgebras agrees with the weaker definition. The claim is posed as a conjecture and is not resolved in the supplied text.