If I bring the first term of expr1 out of Re, Reduce confirms your equality:

Reduce[-((139 \[Pi]^2)/180) + 
   Re[+8 Log[1/2 + Sqrt[5]/2]^2 - 
     3 PolyLog[2, -((-1 + Sqrt[5])/2 )^4] + 
     16 PolyLog[2, ((-1 + Sqrt[5])/2 )^2 E^((I \[Pi])/3)] - 
     2 PolyLog[2, ((-1 + Sqrt[5])/2 )^4 E^((I \[Pi])/3)] + 
     4 PolyLog[2, -((-1 + Sqrt[5])/2 )^2 E^(2 I ArcTan[1/Sqrt[7]])] + 
     8 PolyLog[2, ((-1 + Sqrt[5])/2 )^2 E^(2 I ArcTan[1/Sqrt[7]])] + 
     4 \[Pi] ArcTan[1/Sqrt[7]] - 12 ArcTan[1/Sqrt[7]]^2] == ( 
  2 \[Pi]^2)/15]