Mustăță–Nakamura bounded blow-up conjecture for minimal log discrepancies

Fix a positive integer $N$ and a real exponent $e$. For a smooth $N$-dimensional variety $A$, a real ideal ${\frak{a}}$ with exponent $e$ determines the minimal log discrepancy ${\rm{mld}}(0;A,{\frak{a}})$ at the specified point $0$. A usual blow-up means a blow-up whose center is an irreducible reduced closed subset. Let $\ell_{N,e}\in{\mathbb N}$ denote a bound depending only on $N$ and $e$.

Mustăță–Nakamura conjecture. Fix $N$ and the exponent $e$ of real ideals. Then, there exists a number $\ell_{N,e}\in{\mathbb N}$ depending only on $N$ and $e$ such that for any real ideal ${\frak{a}}$ with the exponent $e$ the minimal log discrepancy ${\rm{mld}}(0;A,{\frak{a}})$ is computed by a prime divisor obtained by at most $\ell_{N,e}$ times blow-ups.

This is the Mustăță–Nakamura conjecture, which motivates the weighted blow-up conjectures discussed in the paper. It asks for a uniform bound on the number of ordinary blow-ups needed to find a divisor computing the minimal log discrepancy, and its resolution status is not specified in the supplied text.