Ten points in a square

It is a standard exercise on Pigeonhole Principle to prove:

If you place 10 distinct points in a square of side 1, then at least
two points will have distance no more than sqrt(2)/3 (about 0.4714).

My question: This number is an upper bound for the minimum
positive distance. Has anyone found the least upper bound?

(It is at least 1/3, just place the points at lattice points
with stepsize 1/3. With slightly more effort, one can replace
1/3 by sqrt(2)/(2*sqrt(2)+1), about 0.3694.)

By the way, tens of millions of pseudorandom experiments have
not exceeded 0.32.

Cheers, ZVK(Slavek).
.

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