Dividing an equilateral triangle into N equal (possibly non-connected) parts

Recently Pavel Guzenko found a way to divide an equilateral triangle into 15 congruent parts (and also into 30 congruent parts).

In a recent preprint https://arxiv.org/abs/1812.07014 M.Beeson shows how to divide an equilateral triangle into $15×3^6=10935$ equal triangles (with sides 3, 5, 7 and one angle equal to $2\pi/3$).

In this MathOverflow thread, there are dissections with $5n^2$ pieces for all $n\ge 6$ where the pieces are simply connected quadrilaterals. The smallest such is pictured here:

In the thread, it is mentioned that Michael Reid claims to have found a simply-connected dissection using $7n^2$ trapezoids for some $n$, but the result appears not to be published anywhere.

In the case where the tiles are triangles, the 1995 paper Tilings of Triangles by M. Laczkovich has many important results. In particular, it states that there is a dissection of an equilateral triangle into $2469600=2^5\cdot3^2\cdot5^2\cdot 7^3$ triangles with side lengths $7, 8,$ and $13$.

In general, Theorem 3.3 in the paper states

Let $x$ and $y$ be non-zero integers such that $x+2y\neq 0\neq y+2x$. Then there is a positive integer $k$ such that the equilateral triangle can be dissected into $n=|xy(x+2y)(y+2x)k^2|$ congruent triangles.

This yields dissections with a number of triangles whose squarefree part is any of $ 5, 6, 10, 13, 14, 15, 21, 30, 35, 39, 55, 65, 66, 70, 85, 95, 105, 119, 130,\ldots$