mg.metric geometry - Smallest dilation of a quadrilateral?

What is the smallest dilation of a quadrilateral in $mathbb{R}^d$?
This may be an open problem, which I know is verboten on MO.
So my question is: Is this indeed open?

It will take me some time to explain the terms.
The notion of dilation derives from Gromov, as far as I know
(He defines a version in
Metric Structures for Riemannian and Non-Riemannian Spaces,
p.11,
although he called it distortion).
I came upon it myself via $t$-spanners.

The version in which I am interested is this.
Let $P$ be a polygon (its boundary, not its interior), and $x,y$ two points on
$P$. You can think of $P$ in $mathbb{R}^2$, but also $mathbb{R}^3$ and $mathbb{R}^d$
for $d>3$ are interesting.
Define $delta(x,y)$ as the maximum (supremum) of $d_P(x,y) / | x y |$,
where $d_P(x,y)$ is the distance between $x$ and $y$ following
$P$ (the shortest path staying on the closed path that consitutes $P$),
and $|xy|$ is the Euclidean distance in $mathbb{R}^d$.
Thus $delta(x,y)$ measures how much $P$ dilates w.r.t. Euclidean distance.
I am interested in the minimum value $delta(P)$
of $delta(x,y)$ over all
$x,y in P$, for all $n$-gons $P$, for fixed $n$.

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Example 1.
If $P$ is a unit square, then $delta(x,y)$
for $x,y$ opposite corners is $sqrt{2}$, but
$delta(P)=2$ because with $x,y$ midpoints of
opposite sides, $delta(x,y)= 2/1$.

Example 2.
If $P$ is an equilateral triangle, $delta(P)=2$, as shown in the figure.
In fact, the dilation of any triangle is $ge 2$ [Lemma 7 in the 2nd paper below].

Example 3.
It is known the the dilation of any closed curve $C$
satisfies $delta(C) ge pi/2$, with equality achieved
only by the circle. [Corollary 23 in the first paper below.] This is (apparently) due to Gromov.

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So I finally come to my question. By reading these two
papers,
"Geometric Dilation of Closed Planar Curves: New Lower Bounds,"
and
"On Geometric Dilation and Halving Chords,"
it appears to me that the minimum dilation of a quadrilateral
in $mathbb{R}^2$ (and $mathbb{R}^d$) is not known.
I had heard this was the case three years ago in a seminar
in Brussels, but (a) I didn't quite believe it,
(b) it was hearsay, and (c) it is now out of date.
I am trying to clarify with the authors of these papers, but in parallel
I would appreciate any information on the status of
this question.
The latter paper cited above proves a lower bound of $4 tan(pi/8) approx 1.66$
(if I have interpreted it correctly).

Finally, if indeed open, this seems a potential
PolyMath undergrad project, as well as fun for anyone else!

Addendum. I don't want to close-out this question, but I have heard from one of the authors of
the above cited papers, and indeed it appears that the dilation of a planar quadrilateral is unknown.
So I have tentatively tagged this as an open-problem, and I will update if new information surfaces. Thanks for everyone's interest and input!