Triangle-roundedness in matroids

A matroid $N$ is said to be triangle-rounded in a class of matroids $\mathcal{M}$ if each $3$-connected matroid $M\in \mathcal{M}$ with a triangle $T$ and an $N$-minor has an $N$-minor with $T$ as triangle. Reid gave a result useful to identify such matroids as stated next: suppose that $M$ is a binary $3$-connected matroid with a $3$-connected minor $N$, $T$ is a triangle of $M$ and $e\in T\cap E(N)$; then $M$ has a $3$-connected minor $M^{\mathrm{\prime }}$ with an $N$-minor such that $T$ is a triangle of $M^{\mathrm{\prime }}$ and $\mathrm{\mid }E(M^{\mathrm{\prime }})\mathrm{\mid }\le \mathrm{\mid }E(N)\mathrm{\mid }+2$. We strengthen this result by dropping the condition that such element $e$ exists and proving that there is a $3$-connected minor $M^{\mathrm{\prime }}$ of $M$ with an $N$-minor $N^{\mathrm{\prime }}$ such that $T$ is a triangle of $M^{\mathrm{\prime }}$ and $E(M^{\mathrm{\prime }})-E(N^{\mathrm{\prime }})\subseteq T$. This result is extended to the non-binary case and, as an application, we prove that $M(K_5)$ is triangle-rounded in the class of the regular matroids.