If $p$ is both Giuga and Carmichael number

then its known that

$1^{p-1}+2^{p-1}+3^{p-1}+\cdots+(p-1)^{p-1} \equiv -1\pmod{p}$

is it true that

if $p$ is both Giuga and Carmichael number then

$1^{p-1}+2^{p-1}+3^{p-1}+\cdots+(r-1)^{p-1} \equiv (r-1)\pmod{p}$ where $2\le r\le p-2$

Thanks in advance :)