Well-definedness and analytic behavior of the multiplicative Euler characteristic

Let $\rho$ and $\pi$ be admissible representations in the local-field setting, and let

$$\cdots \longrightarrow M_i \longrightarrow M_{i-1} \longrightarrow \cdots \longrightarrow M_0 \longrightarrow V \longrightarrow 0$$

be a monomial resolution, with each $L(s,M_i,\pi)$ the product of the two-variable $L$-functions associated with the monomial summands of $M_i$. The multiplicative Euler characteristic conjecture. The multiplicative Euler characteristic of the two-variable $L$-functions $L(s,M_i,\pi)$ defines a well-defined and analytically well-behaved $L$-function $L(s,\rho,\pi)$. In the di-$p$-adic situation the monomial resolution need not be of finite type, so the conjecture asks that the resulting multiplicative Euler characteristic nevertheless has the expected well-definedness and analytic properties.