Relative Hilali conjecture for maps

Let $f:X\to Y$ be a continuous map between simply connected elliptic spaces. Let

$$P_f(t)=1+\sum_{i\geq 2}t^i\dim\operatorname{Ker}H_i(f;\mathbb Q),$$

and

$$P_f^{\pi}(t)=\sum_{i\geq 2}t^i\dim\operatorname{Ker}(\pi_i(f)\otimes\mathbb Q).$$

Relative Hilali conjecture. The inequality

$$P_f^{\pi}(1)\leq P_f(1),$$

that is,

$$\sum_{i\geq 2}\dim\operatorname{Ker}(\pi_i(f)\otimes\mathbb Q)\leq 1+\sum_{i\geq 2}\dim\operatorname{Ker}H_i(f;\mathbb Q),$$

should hold. The conjecture was proposed in earlier work and has been proved in some cases; its general validity remains unresolved in the stated source.