Let $M^{2n}$ be a smooth orientable manifold. The tangent bundle $TM$ of $M$ is induced by a classifying map $f:M\to BSO(2n)$ which is unique up to homotopy. An almost complex structure on $M$ is a homotopy class $[\tilde{f}]$ of a lift $\tilde{f}:M\to BU(n)$ of $f$ to $BU(n)$.
Let $\tilde{f}_1,\tilde{f}_2\in [\tilde{f}]$ be two lifts in the same homotopy class and $F:[0,1]\times M \to BU(n)$ a homotopy from $\tilde{f}_1$ to $\tilde{f}_2$.
Question: Is $F(t,\cdot):M\to BU(n)$ an almost complex structure on $M$ for all $t\in [0,1]$?