  We conjecture that $W$ gravity can be interpreted as the gauge theory of
$\phi$ diffeomorphisms in the space of dimensionally-reduced $D=2+2$
$SU^*(\infty)$ Yang-Mills instantons. These $\phi$ diffeomorphisms preserve a
volume-three form and are those which furnish the correspondence between the
dimensionally-reduced Plebanski equation and the KP equation in $(1+2)$
dimensions. A supersymmetric extension furnishes super-$W$ gravity. The
Super-Plebanski equation generates self-dual complexified super gravitational
backgrounds (SDSG) in terms of the super-Plebanski second heavenly form. Since
the latter equation yields $N=1~D=4~SDSG$ complexified backgrounds associated
with the complexified-cotangent space of the Riemannian surface,
$(T^*\Sigma)^c$, required in the formulation of $SU^*(\infty)$ complexified
Self-Dual Yang-Mills theory, (SDYM ); it naturally follows that the recently
constructed $D=2+2~N=4$ SDSYM theory- as the consistent background of the open
$N=2$ superstring- can be embedded into the $N=1~SU^*(\infty)$ complexified
Self-Dual-Super-Yang-Mills (SDSYM) in $D=3+3$ dimensions. This is achieved
after using a generalization of self-duality for $D>4$. We finally comment on
the the plausible relationship between the geometry of $N=2$ strings and the
moduli of $SU^*(\infty)$ complexified SDSYM in $3+3$ dimensions.
