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Heged\\\"{u}s and Frankl recently extended the concept of Bollob\\'as systems to $d$-tuples, conjecturing that for a Bollob\\'as system of $d$-tuples, $\\{(A_i^{(1)},\\ldots,A_i^{(d)})\\mid i\\in[m]\\}$, the maximum value of $\\sum_{i=1}^m\\binom{|A_i^{(1)}|+\\cdots+|A_i^{(d)}|}{|A_i^{(1)}|,\\ldots,|A_i^{(d)}|}^{-1}$ is also $1$. This paper refutes this conjecture and establishes an upper bound for the sum. 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Heged\\\"{u}s and Frankl recently extended the concept of Bollob\\'as systems to $d$-tuples, conjecturing that for a Bollob\\'as system of $d$-tuples, $\\{(A_i^{(1)},\\ldots,A_i^{(d)})\\mid i\\in[m]\\}$, the maximum value of $\\sum_{i=1}^m\\binom{|A_i^{(1)}|+\\cdots+|A_i^{(d)}|}{|A_i^{(1)}|,\\ldots,|A_i^{(d)}|}^{-1}$ is also $1$. This paper refutes this conjecture and establishes an upper bound for the sum. 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