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Imai, M. Kato, H. Takamura, K. Wakasa, The lifespan of solutions of semilinear wave equations with the scale-invariant damping in two space dimensions, J. Differential Equations 269 (2020), no. 10, 8387-8424], it is conjectured that the global small data weak solution $u$ exists when $p>p_{s}(2+\\mu) =\\frac{\\mu+3+\\sqrt{\\mu^2+14\\mu+17}}{2(\\mu+1)}$ for $\\mu\\in (0, 2)$ and $p>p_f(2)=2$ for $\\mu\\geq 2$. 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