Conjecture on the Euler characteristic of finite CW-complexes

Let $X$ be a simply-connected finite CW-complex, let ${\Bbbk}$ be a field, and let $i:\Omega X\rightarrow LX$ be the inclusion of based loops into free loops. Suppose that

$$H^*(i;{\Bbbk}):H^*(LX;{\Bbbk})\longrightarrow H^*(\Omega X;{\Bbbk})$$

is onto. The conjecture. Then $\chi(X)$ is zero modulo the characteristic of ${\Bbbk}$, or $H_*(X)\cong {\Bbbk}$. This predicts a restriction on the Euler characteristic, or homology, of $X$ under surjectivity of the cohomology map induced by the inclusion of based loops. The source gives no resolution of the conjecture.