Equilaterally triangulated surfaces with prescribed boundary

We resolve Kenyon's problem in this paper. We discuss in Section 5 a number of further conjectures and open problems.

(It is intended as an extended comment and thinking out loud rather than an answer)

That seems that the case when $p$ is a quadrilateral is exactly the bottleneck, the general case will hopefully follow from it by an induction argument.

Without loss of generality we can prove the statement for paths of even length (glue a triangle to any edge of an odd-length path and span the resulting even-length path).

Definition: The waist $w(p)$ of a closed polygonal path $p$ of lenght $l$ is the shortest line segment connecting two vertices of $p$ that separate $p$ into pieces of equal lenght ($=\frac{l}{2}$)

Statement: $|w(p)|\le C\cdot l$, where $C$ is the ratio of the diameter of the regular polygon with $l$ edges of length 1 to its perimeter $l$.

In other words, a waist of a path of length $l$ cannot be longer than a diagonal of the regular $l$-gon. In particular it is quite short for large $l$, shorter than $\frac{l}{2}$.

We can turn the waist into a polygonal path (with unit segments) of length $\lceil{|w|}\rceil$ (smallest integer greater than $|w|$). Starting from $l=6$, when $C=\frac{1}{3}$ and a poligonized waist (let's call it $W$) has lenght at worst $2$ -- everything is more or less fine: span the path consisting of the first half of $p$ and $W$ (its length at worst $5$ -- 5 is not even, of course, but at least less than 6 :). For $l=8$ it's surely curable) and in the same manner span the second half of $p$ and $W$ and take the union of these two spanning surfaces, doing an induction step.

For $l=4$ that does't work, however, since $W$ can have length $2$.

Even for a transcendentally angled flat rhombus it is not clear to me whether one can span it. If no, the answer can be probably given by some sort of extension of fields argument (in any construction I tried to think of the transcendence of the angle survived in some form preventing to complete it). In the positive direction, there is the industry of flexible polyhedra, which potentially could provide an example of a polyhedra with movable quadrilateral living on it that could be a hope for a construction, but I could not readily find in the literature ones with equilateral triangle sides with enough flexibility to do something like we want.