  We study various duality webs involving the 3d FT[SU(N)] theory, a close
relative of the T[SU(N)] quiver tail. We first map the partition functions of
FT[SU(N)] and its 3d spectral dual to a pair of spectral dual q-Toda conformal
blocks. Then we show how to obtain the FT[SU(N)] partition function by Higgsing
a 5d linear quiver gauge theory, or equivalently from the refined topological
string partition function on a certain toric Calabi-Yau three-fold. 3d spectral
duality in this context descends from 5d spectral duality. Finally we discuss
the 2d reduction of the 3d spectral dual pair and study the corresponding
limits on the q-Toda side. In particular we obtain a new direct map between the
partition function of the 2d FT[SU(N)] GLSM and an (N+2)-point Toda conformal
block.
