I am looking for some result useful in deriving following (conjecture?):

Let $A$ be an $n\times n$ matrix with $0-1$ entries. Suppose, that exactly $k\leqslant n$ entries are equal to $0$. Then $\mathrm{Per} A \leqslant n!\left(1-\frac{k}{2n}\right)$

Unfortunately, trying to derive this inequality from other estimates fails in general. For example, the inequality given in

Adam G. Weyhaupt,

A note on some upper bounds for permanents of (0,1)-matrices, Journal of Interdisciplinary Mathematics12no 1 (2009) pp 123–128, doi:10.1080/09720502.2009.10700615 (pdf)

failed with $k=4$.