Grigoriev's effective tropical dual Nullstellensatz conjecture

Let $F=\{f_1,\ldots,f_k\}$ be a system of tropical polynomials in $n$ variables, and let $M_N$ be the finite Macaulay matrix obtained from the monomial multiples of the $f_i$ of degree at most $N$. A tropical root is a common point at which each polynomial in $F$ attains its minimum at least twice. Grigoriev's effective tropical dual Nullstellensatz conjecture. There is a function $N$ of $n$ and of $\deg(f_i)$ for $1\leq i\leq k$ such that $F$ has a common tropical root if and only if the tropical linear system corresponding to $M_N$ has a solution. The conjecture gives an effective dual formulation of the tropical Nullstellensatz: solvability of the tropical linear system should detect common tropical roots with a degree bound depending only on the number of variables and the input degrees. It was known in the cited work for polynomials in one variable; the general effective bound remains open.