Hu'ng's weak algebraic conjecture on spherical classes

Let $A$ be the mod-$2$ Steenrod algebra, let $D_k$ be the Dickson algebra, and let

$$\varphi_k:\operatorname{Ext}^{k,k+i}_A(\mathbb Z/2,\mathbb Z/2)\to (\mathbb Z/2\otimes D_k)^*_i$$

be the Lannes–Zarati homomorphism. Let

$$Tr_k:\mathbb Z/2\otimes_{GL_k}PH_i(BV_k)\to \operatorname{Ext}^{k,k+i}_A(\mathbb Z/2,\mathbb Z/2)$$

be Singer's algebraic transfer, where $V_k=(\mathbb Z/2)^{\times k}$ and $P(-)$ denotes the primitive submodule functor. Weak algebraic conjecture on spherical classes. For $k>2$, the composition

$$\varphi_k\circ Tr_k:\mathbb Z/2\otimes_{GL_k}PH_i(BV_k)\to (\mathbb Z/2\otimes D_k)^*_i$$

is trivial.

This weak form asks only for vanishing after applying Singer's algebraic transfer, so it is weaker than the algebraic conjecture on spherical classes. The supplied text does not state a resolution, so its status is left open.