  We investigate the conjectured infinite-dimensional hidden symmetries of
six-dimensional chiral supergravity coupled to two vector multiplets and two
tensor multiplets, which is known to possess the $F_{4,4}$ symmetry upon
dimensional reduction to three spacetime dimensions. Two things are done. (i)
First, we analyze the geodesic equations on the coset space
$F_{4,4}^{++}/K(F_{4,4}^{++})$ using the level decomposition associated with
the subalgebra $\mathfrak{gl}(5)\oplus \mathfrak{sl}(2)$ of $F_{4,4}^{++}$ and
show their equivalence with the bosonic equations of motion of six-dimensional
chiral supergravity up to the level where the dual graviton appears. In
particular, the self-duality condition on the chiral $2$-form is automatically
implemented in the sense that no dual potential appears for that $2$-form, in
contradistinction with what occurs for the non chiral $p$-forms. (ii) Second,
we describe the $p$-form hierarchy of the model in terms of its $V$-duality
Borcherds superalgebra, of which we compute the Cartan matrix.
