Here is the beginning of an attempt. On the first weighing, put 123 v 456. On the second weighing, put 147 v 258. Now the possibilities remaining in each case, where I write “Even” if the scales are balanced and otherwise write the name of the heavier side, are:

Even Even: +15, -14, +24, -24, +36, -36, +78, -78

Even Left: +14, +16, +34, -25, -26, -35

Even Right: -14, -16, -34, +25, +26, +35

Left Even: +12, +18, +27, -45, -48, -57

Left Left: +13, +17, +37, -46, -58, -68

Left Right: +23, +28, +38, -46, -47, -67

Right Even: -12, -18, -27, +45, +48, +57

Right Left: -13, -17, -37, +46, +58, +68

Right Right: -23, -28, -38, +46, +47, +67

So after two weighings there are only 6 or 8 possibilities remaining. I leave it to others to show that each case can or cannot be solved in 2 weighings.