Smallest singular value bound for general deterministic shifts of real Ginibre matrices

Let $X$ be an $N\times N$ real Ginibre matrix, let $A$ be a deterministic matrix, let $\gamma>0$, and define the Hermitian imaginary part by

$$\Im A:=\frac{1}{2\mathrm{i}}(A-A^*).$$

Write $\sigma_1(\gamma X+A)$ for the smallest singular value and let $\operatorname{Tr}(\Im A)^2$ denote the trace of $(\Im A)^2$. General shifted real Ginibre smallest singular value conjecture. There exist constants $c_*,C_*>0$ such that

$$\operatorname{\mathbf{P}}\left(\sigma_1(\gamma X+A)\le \frac{\sqrt{x}}{N}\right)\le C_*\frac{x}{\gamma^2}+C_*e^{-c_*\operatorname{Tr}(\Im A)^2}\left(\frac{\sqrt{x}}{\gamma}\wedge\frac{x}{\sqrt{\operatorname{Tr}(\Im A)^2}\gamma^2}\right).$$

The conjecture extends the shifted case $A=-zI$ to arbitrary deterministic matrices; the source notes that its preceding theorem proves the special case up to a logarithmic correction. The source does not state a resolution beyond this partial result.