Let $A$ be a doubly stochastic matrix of order $n$
and $\sigma_i(A)$ the sum of the order $i$ subpermanents of $A$. 
The HD conjecture says $in\sigma_i(A)\geq(n-i+1)^2\sigma_{i-1}(A)$ for
each $i=1,2,\dots,n$.  There is a natural counterpart of this conjecture
for $(0,1)$-matrices with constant line sum $k$.  We show this
associated conjecture holds when $k\leq2$, $k\geq n-2$,
$i\leq n/k+1$ or $i\leq5$ but that it fails in general because it
(wrongly) asserts a polynomial bound on the ratio of near perfect to
perfect matchings in $k$-regular bipartite graphs.