  I show that a hypergeometric function
${}_{p}F_q(a_1,\ldots,a_p;b_1,\ldots,b_q;\,\cdot\,)$ with $p \le q$ belongs to
the Laguerre--P\'olya class $LP^+$ for arbitrarily large $b_{p+1},\ldots,b_q >
0$ if and only if, after a possible reordering, the differences $a_i - b_i$ are
nonnegative integers. This result arises as an easy corollary of the case $p=q$
proven two decades ago by Ki and Kim. I also give explicit examples for the
case ${}_{1}F_2$.
