recreational mathematics - Walking to infinity on the primes: The prime-spiral moat problem

It is an unsolved problem to decide if it is possible to "walk to infinity" from the origin
with bounded-length steps, each touching a Gaussian prime as a stepping stone.
The paper by Ellen Gethner, Stan Wagon, and Brian Wick,
"A Stroll through the Gaussian Primes"
(American Mathematical Monthly, 105: 327-337 (1998))
discusses this Gaussian moat problem and proves that steps of length $< sqrt{26}$ are
insufficient. Their result was improved to $sqrt{36}$ in 2005.

My question is:

Is the analogous question easier for the
prime spiral (a.k.a. Ulam spiral)—Can one walk to infinity using bounded-length steps
touching only the spiral coordinates of primes?

What little I know of prime gaps
suggest that should be easier to walk to infinity.
For example, the first gap of 500 does not occur until about $10^{12}$
(more precisely, 499 and 303,371,455,241).

I ask this primarily out of curiosity, and have tagged it 'recreational.'

Edit1. In light of Gjergji's remarks below, I have tagged this as an open problem.

Edit2'.
Just for fun, I computed which primes are reachable on a small portion of the spiral,
for step distances $d le 3$ (left below) and $d le 4$ (right below);
red=reached, blue=not reached.
The former does not reach 83, the 23^rd prime blue dot barely discernable at spiral coordinates (5,-3);
the latter does not reach 5087, the 680^th prime blue dot at
spiral coordinates (36,10).

• Next fa.functional analysis - Spectral theory for self-adjoint field operators on a symmetric Fock space
• Previous high energy astrophysics - meaning of p-wave charmonia