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For a function $F:\{0,1\}^n\times\{0,1\}^n\rightarrow\{0,1\}$, distribution $\mu$ on inputs $\{0,1\}^n\times\{0,1\}^n$, where Alice's and Bob's inputs are random variables $X$ and $Y$, respectively, the external information complexity for two-player zero-error protocols is defined as
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For a function $F:\{0,1\}^n\times\{0,1\}^n\rightarrow\{0,1\}$, distribution $\mu$ on inputs $\{0,1\}^n\times\{0,1\}^n$, where Alice's and Bob's inputs are random variables $X$ and $Y$, respectively, external information complexity for two-player, zero-error protocols is defined as follows.
 
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\textrm{IC}^\text{ext}(F,0,\mu) := \inf_{\Pi \text{ that solve $F$ correctly always}} I_\mu(\Pi;XY)\,.
 
\textrm{IC}^\text{ext}(F,0,\mu) := \inf_{\Pi \text{ that solve $F$ correctly always}} I_\mu(\Pi;XY)\,.
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We denote by $\overline{\textrm{CC}}(F^n,0,\mu^n)$ the expected communication complexity of $F^n$ with respect to the distribution $\mu^n$ for zero-error protocols.
 
We denote by $\overline{\textrm{CC}}(F^n,0,\mu^n)$ the expected communication complexity of $F^n$ with respect to the distribution $\mu^n$ for zero-error protocols.
  
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Either prove or disprove that
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Either prove or disprove the following conjecture.
 
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\textrm{IC}^\text{ext}(F,0,\mu) = \lim_{n\rightarrow\infty} \frac{\overline{\textrm{CC}}(F^n,0,\mu^n)}{n}\,.
 
\textrm{IC}^\text{ext}(F,0,\mu) = \lim_{n\rightarrow\infty} \frac{\overline{\textrm{CC}}(F^n,0,\mu^n)}{n}\,.

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