Prime spiral distribution into quadrants

Is it known that the primes on the Ulam prime spiral distribute themselves equally in sectors around the origin? To be specific, say the quadrants? (Each quadrant is closed on one axis and open on the other.) For example, in the $50 \times 50$ spiral below, I count the number of primes in the four quadrants to be $(103,96,88,86)$ ($\sum=373$), leading to ratios $(0.276,0.257,0.236,0.231)$:
     
For the $500 \times 500$ spiral, I count $(5520,5553,5535,5469)$ ($\sum=22077$) leading to $(0.250,0.252,0.251,0.248)$. Empirically there is a convergence to $\frac{1}{4}$, but I wonder if this has been proven?

Thanks!

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An affirmative answer has been given on MathOverflow.