Existence of homotopically non-trivial inclusion map from $X\simeq \mathbb{S}^6$ to $Y\simeq \mathbb{S}^4\vee \mathbb{S}^7$

Question

Let $X$ be a subcomplex of a simplicial complex $Y$ such that $X\simeq \mathbb{S}^6$ and $Y\simeq \mathbb{S}^7\vee\mathbb{S}^4$.

Question: Is the inclusion map $i :X \longrightarrow Y$ always null homotopic?

Any ideas or references are greatly appreciated!