Perform an explicit summation, if possible, over a restricted range of two indices both varying from 0 to infinity

Question

I would like to sum the term

 P=(32 (1/(1 - w))^k w^(-3 + h) ((8 - 3 w) w + 2 h (-4 + w) (-4 + 3 w)) Gamma[-(1/2) + h]*
   Gamma[1/2 + h])/((1 + 2 k)^2 (-3 + 4 k (1 + k)) π^3 (1 - w)^(3/4) Pochhammer[1, h]^2)

over both $h$ and $k$ from 0 to $\infty$, but OMIT precisely the range for which the conditions $h > 2$ and $h < 3/4 + k$ simultaneously hold. (The range to be omitted is that for which the integral of P can be directly computed--so an initial summation can be avoided in that case.) Then, the numerical integral of the result over $w \in [-\infty,0]$ should give 0.25795729571462833563875652. However, we would hope to compute the integral exactly.

I'm not sure how best to set up this restricted summation problem in Mathematica (not too optimistic about it being, in fact, doable).

This is an auxiliary question to Sum a certain hypergeometric-function-based expression pertaining to an integration problem

The unrestricted summation of $P$ over the indices $h,k$ from 0 to $\infty$ is given by the product of

1/(π^3 (1 - w)^(1/4) (-1 + w) w^3)

and

   8 (-16 EllipticE[w] + 8 w EllipticE[w] + 16 EllipticK[w] - 
   16 w EllipticK[w] + 3 w^2 EllipticK[w]) (Sqrt[1 - w] w - 
   2 w ArcTanh[1/Sqrt[1 - w]] + w^2 ArcTanh[1/Sqrt[1 - w]] + 
   Sqrt[1 - w] LerchPhi[1/(1 - w), 2, 1/2])

Its (numerical) integral over $w \in [-\infty,0]$ gives the "operator monotone $\sqrt{x}$ two-rebit separability probability" shown by Lovas and Andai (https://arxiv.org/abs/1610.01410 p. 16) to be $\approx 0.26223$. It's an exact underlying value for this term that I've been seeking to find here and in the indicated auxiliary post Sum a certain hypergeometric-function-based expression pertaining to an integration problem .

Answer 1

So for the complete sum, the integral is available. But for the partial sum, the individual integrals are available but so for the partial sum. Why not just perform a numerical integration on the complete sum with higher precision than what was done before?

Chop[NIntegrate[(1/(π^3 (1 - w)^(1/4) (-1 + w) w^3))*(8 (-16 EllipticE[w] + 8 w EllipticE[w] + 
  16 EllipticK[w] - 16 w EllipticK[w] + 3 w^2 EllipticK[w]) (Sqrt[1 - w] w - 
  2 w ArcTanh[1/Sqrt[1 - w]] + w^2 ArcTanh[1/Sqrt[1 - w]] + 
  Sqrt[1 - w] LerchPhi[1/(1 - w), 2, 1/2])), {w, -∞, 1/100}, 
  MaxRecursion -> 10, WorkingPrecision -> 30]]

(* 0.262222891715939967267906731685 *)

Without using something like MaxRecursion -> 10, WorkingPrecision -> 30 one gets

(* 0.26223 *)