Estimates in the ideal problem for the algebra $H^\infty$

Previous results by the author and S. V. Kislyakov (Algebra i Analiz, 35, No. 5 (2023), 99–116) about the independence of the solvability of the ideal problem on the nature of spaces in which it is posed are extended to incorporate the context of the work by S. R. Treil (J. Funct. Analysis, 253 (2007), 220–240) and J. Pau (Proc. Amer. Math. Soc., 133 (2005), 167–174).