Budur–van der Veer–Van Werde upper-bound problem

Let $X$ be a smooth complex affine variety of dimension $n$, let $F=(f_1,\ldots,f_r)$ be a tuple of nonzero regular functions such that $f=\prod_{i=1}^r f_i$ is not invertible, and let $\mathbf a\in\mathbb Z_{cge 0}^r$. Fix a log resolution of $f$, and for each exceptional or relevant divisor $E$ write $m_{i,E}=\operatorname{ord}_E(f_i)$, $L_E(\mathbf s)=\sum_{i=1}^r m_{i,E}s_i$, $L_E(\mathbf 1)=\sum_{i=1}^r m_{i,E}$, and let $k_E$ denote the coefficient of $E$ in the relative canonical divisor. Every codimension-one irreducible component of $Z(B_F^{\mathbf a})$ has the form $L_E(\mathbf s)+k_E+c=0$ for a positive integer $c$. The problem is to prove, for arbitrary tuples $F$ and shifts $\mathbf a$, the upper bound $c\le L_E(\mathbf a)+(n-1-\delta_f)L_E(\mathbf 1)-k_E$, where $\delta_f=\min\{n-1,\alpha_f\}$ and $\alpha_f$ is the minimal exponent of $f$.