  Ollivier and Lin--Lu--Yau established the theory of graph Ricci curvature
(LLY curvature) via optimal transport on graphs.
Ikeda--Kitabeppu--Takai--Uehara introduced a new distance called the
Kantorovich difference on hypergraphs and generalized the LLY curvature to
hypergraphs (IKTU curvature). As the LLY curvature can be represented by the
graph Laplacian by M\"unch--Wojciechowski, Ikeda--Kitabeppu--Takai--Uehara
conjectured that the IKTU curvature has a similar expression in terms of the
hypergraph Laplacian. In this paper, we introduce a variant of the Kantorovich
difference inspired by the above conjecture and study the Ricci curvature
associated with this distance ($\mathsf{wIKTU}$ curvature). Moreover, for
hypergraphs with a specific structure, we analyze a quantity $\mathcal{C}(x,y)$
at two distinct vertices $x,y$ defined by using the hypergraph Laplacian. If
the resolvent operator converges uniformly to the identity, then
$\mathcal{C}(x,y)$ coincides with the $\mathsf{wIKTU}$ curvature along $x,y$.
