  Recent work explores the candidate phases of the 4d adjoint quantum
chromodynamics (QCD$_4$) with an SU(2) gauge group and two massless adjoint
Weyl fermions. Both Cordova-Dumitrescu and Bi-Senthil propose possible low
energy 4d topological quantum field theories (TQFTs) to saturate the higher 't
Hooft anomalies of adjoint QCD$_4$ under a renormalization-group (RG) flow from
high energy. In this work, we generalize the symmetry-extension method
[arXiv:1705.06728] to higher symmetries, and formulate higher group cohomology
and cobordism theory approach to construct higher-symmetric TQFTs. We prove
that the symmetry-extension method saturates certain anomalies, but also prove
that neither $A \mathcal{P}_2(B_2)$ nor $\mathcal{P}_2(B_2)$ can be fully
trivialized, with the background 1-form field $A$, Pontryagin square
$\mathcal{P}_2$ and 2-form field $B_2$. Surprisingly, this indicates an
obstruction to constructing a fully 1-form center and 0-form chiral symmetry
(full discrete axial symmetry) preserving 4d TQFT with confinement, a no-go
scenario via symmetry-extension for specific higher anomalies. We comment on
the implications and constraints on deconfined quantum critical points (dQCP),
quantum spin liquids (QSL) or quantum fermionic liquids in condensed matter,
and ultraviolet-infrared (UV-IR) duality in 3+1 spacetime dimensions.
