A stronger version of the van der Waerden permanent conjecture asserts that if
J
n
{J_n}
denotes the
n
×
n
n \times n
matrix all of whose entries are
1
/
n
1/n
and
A
A
is any fixed matrix on the boundary of the set of
n
×
n
n \times n
doubly stochastic matrices, then
per
(
λ
A
+
(
1
−
λ
)
J
n
)
{\text {per}}(\lambda A + (1 - \lambda ){J_n})
as a function of
λ
\lambda
is nondecreasing in the interval
[
0
,
1
]
[0,1]
. In this paper, we elucidate the relation of this assertion to some other conjectures known to be stronger than van der Waerden’s. We also show that this assertion is true when
n
=
3
n = 3
and in the case when, up to permutations of rows and columns, either (i)
A
=
J
s
⊕
J
t
A = {J_s} \oplus {J_t}
,
0
>
s
0 > s
,
t
t
,
s
+
t
=
n
s + t = n
or (ii)
A
=
[
0
a
m
p
;
Y
Y
T
a
m
p
;
Z
]
A = \left [\begin {smallmatrix} 0 & Y \\ Y^T & Z\end {smallmatrix} \right ]
where 0 is an
s
×
s
s \times s
zero matrix,
Y
Y
is
s
×
t
s \times t
with all entries equal to
1
/
t
1/t
, and
Z
Z
is
t
×
t
t \times t
with all entries equal to
(
t
−
s
)
/
t
2
(t - s)/{t^2}
,
0
>
s
⩽
t
0 > s \leqslant t
,
s
+
t
=
n
s + t = n
.