The Farmyard Problem: Minimize the amount of ditch-digging needed to locate a straight pipe under a square piece of land

Reposting from the comments so this question will have an answer:

This problem appears to be unsolved. This paper (Dumitrescu, Jiang, and Pach, Opaque Sets, preliminarily released in 2011, now dated May 23, 2018) provides an example with total length $\sqrt{2}+\frac{\sqrt{6}}{2} = 2.638958433764...$ which is conjectured to be optimal. I believe this uses a Steiner tree to connect 3 vertices of the square and then a diagonal line from the 4th vertex to the center of the square. Quoting:

The diagonal segment $[(\frac{1}{2}, \frac{1}{2}),(1, 1)]$ together with three segments connecting the corners $(0,1), (0,0), (1,0)$ to the point $(\frac{1}{2}−\frac{\sqrt{3}}{6},\frac{1}{2}−\frac{\sqrt{3}}{6})$ yield a barrier of length $\sqrt{2}+\frac{\sqrt{6}}{2}$ =2.639

This is illustrated by this drawing derived from https://www.susqu.edu/brakke/opaque/opaqsq.html

The paper R.E.D. Jones, Opaque sets of degree α, American Mathematical Monthly, May 1964, p. 535-537, establishes a lower bound of 2.