  A famous theorem of Kalai and Meshulam is that $\mathrm{reg}(I + J) \leq
\mathrm{reg}(I) + \mathrm{reg}(J) -1$ for any squarefree monomial ideals $I$
and $J$. This result was subsequently extended by Herzog to the case where $I$
and $J$ are any monomial ideals. In this paper we conjecture that the
Castelnuovo--Mumford regularity is subadditive on binomial edge ideals.
Specifically, we propose that $\mathrm{reg}(J_{G}) \leq \mathrm{reg}(J_{H_{1}})
+ \mathrm{reg}(J_{H_{2}}) -1$ whenever $G$, $H_{1}$, and $H_{2}$ are graphs
satisfying $E(G) = E(H_{1}) \cup E(H_{2})$ and $J_{\ast}$ is the associated
binomial edge ideal. We prove a special case of this conjecture which
strengthens the celebrated theorem of Malayeri--Madani--Kiani that
$\mathrm{reg}(J_{G})$ is bounded above by the minimal number of maximal cliques
covering the edges of the graph $G$. From this special case we obtain a new
upper bound for $\mathrm{reg}(J_{G})$, namely that $\mathrm{reg}(J_{G}) \leq
\mathrm{ht}(J_{G}) +1$. Our upper bound gives an analogue of the well-known
result that $\mathrm{reg}(I(G)) \leq \mathrm{ht}(I(G)) +1$ where $I(G)$ is the
edge ideal of the graph $G$. We additionally prove that this conjecture holds
for graphs admitting a combinatorial description for its Castelnuovo--Mumford
regularity, that is for closed graphs, bipartite graphs with $J_{G}$
Cohen--Macaulay, and block graphs. Finally, we give examples to show that our
new upper bound is incomparable with Malayeri--Madani--Kiani's upper bound for
$\mathrm{reg}(J_{G})$ given by the size of a maximal clique disjoint set of
edges.
