A Strict Version of the Non-commutative Urysohn Lemma

Given a pair $B$, $C$ of $q$-commuting, hereditary $C^{*}$-subalgebras of a unital $C^{*}$-algebra $A$, such that $B\cap C$ is $\sigma$-unital and $1\in B + C$, there is an element $h$ in $A$, with $0\le h\le 1$, such that $h$ is strictly positive in $B$ and $1 - h$ is strictly positive in in $C$. Moreover, $h - h^{2}$ is strictly positive in in $B\cap C$.