Several things can cause shimmy at that speed.

Wheels out of balance (by far the cheapest solution if it is indeed the problem).

Old tires with tread seperation issues (usually accompanied by premature inner tread wear).

Tires that are out of spec with factory requirements for the OEM rims (usually a too wide tire will cause this)

Worn control arm bushings or ball joint wear (not unusual for a car your age with a healthy amount of mileage).

If you rims are not OEM, some are manufactured with defects such as eccentric hubs (and corresponding bolt holes). While the wheel may look fine statically, even a minute tolerance offset will manifest itself at high speeds.

As already suggested, swap the front sets with the rear to see if the problem goes away. Then try a DYNAMIC wheel balance next.

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2009 ML350 (106K) - Family vehicle
2001 CLK430 Cabriolet (80K) - Wife's car
2005 BMW 645CI (138K) - My daily driver
2016 Mustang (32K) - Daughter's car