  The ultrashort unitary (4,0) supermultiplet of 6d superconformal algebra
OSp(8*|8) reduces to the CPT-self conjugate supermultiplet of 4d superconformal
algebra SU(2,2|8) that represents the fields of maximal N=8 supergravity. The
graviton in the (4,0) multiplet is described by a mixed tensor gauge field
which can not be identified with the standard metric in 6d. Furthermore the
(4,0) supermultiplet can be obtained as a double copy of (2,0) conformal
supermultiplet whose interacting theories are non-Lagrangian. It had been
suggested that an interacting non-metric (4,0) supergravity theory might
describe the strongly coupled phase of 5d maximal supergravity. In this paper
we study the implications of the existence of an interacting non-metric (4,0)
supergravity in 6d. The (4,0) theory can be truncated to non-metric (1,0)
supergravity coupled to 5,8 and 14 self-dual tensor multiplets that reduce to
three of the unified magical supergravity theories in d=5. This implies that
the three infinite families of unified N=2 , 5d Maxwell-Einstein supergravity
theories (MESGTs) plus two sporadic ones must have uplifts to unified
non-metric (1,0) tensor Einstein supergravity theories in d=6. These theories
have non-compact global symmetry groups under which all the self-dual tensor
fields including the gravitensor transform irreducibly. Four of these theories
are uplifts of the magical supergravity theories whose scalar manifolds are
symmetric spaces. The scalar manifolds of the other unified theories are not
homogeneous spaces. We also discuss the exceptional field theoretic
formulations of non-metric unified $(1,0)$ tensor-Einstein supergravity
theories and conclude with speculations concerning the existence of higher
dimensional non-metric supergravity theories that reduce to the $(4,0)$ theory
in $d=6$.
