As established in a previous post Equilateral Triangles In The Taxicab Space there are exactly $8$ equilateral triangles (of edge length $1$) that can be packed in the $l^1(\mathbb{R}^2)$ unit circle, while there are only $6$ equilateral triangles (of edge length $1$) that can be packed in the $l^2(\mathbb{R}^2)$ unit circle.
My suspicion is that if one were to consider an $l^{1 + \epsilon}(\mathbb{R}^2)$ unit circle, then for sufficiently small epsilon, the number of equilateral triangles which can be packed in the unit circle is still $8$. More so, I suspect that for a specific value of $\epsilon >0$ the $l^{1+ \epsilon}(\mathbb{R}^2)$ unit circle can only contain $7$ equilateral triangles.
Question 1: Is my suspicion correct?
Question 2: Is there a value $\epsilon >0$ such that the $l^{1+ \epsilon}(\mathbb{R}^2)$ unit circle contains exactly $7$ equilateral triangles?