Infinite-volume composition conjecture for tg-hyperbolic virtual knots and links

Let $K_1$ and $K_2$ be tg-hyperbolic virtual knots or links with genera $g_1,g_2>0$. For corks $C_1^i$ and $C_2$, form compositions

$$W_i=(S_1\times I,K_1,C_1^i)\mathbin{\#_{ns}}(S_2\times I,K_2,C_2).$$

Infinite-volume composition conjecture. There exists a sequence of compositions $W_i$ such that

$$\lim_{i\to\infty}\operatorname{vol}(W_i)=\infty.$$

The surrounding discussion presents this as an expected extension of the alternating case and notes that, if true, it would give infinitely many compositions with volumes approaching infinity for any two tg-hyperbolic virtual knots.