Let $X = Q_1\cap Q_2$ be a complete intersection of two smooth quadrics, over a field $K$, in $\mathbb{P}^4$ with homogeneous coordinates $y_0,y_1,y_2,y_3,y_4$.

Set $Q_1 = \{F_1 = 0\}$ and $Q_2 = \{F_2 = 0\}$ and assume that the monomial $y_0^2$ does not appear in $F_1$ so that $Q_1$ is rational and that the monimial $y_0y_1$ does not appear in $F_2$.

Under these hypotheses could we conclude anything about the unirationality over $K$ of $X$?

What if $X$ is a complete intersection of two quadrics in $\mathbb{P}^n$ with the same properties for $n\geq 5$?