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When setting multiple values for a checkox property, the values should be separated with a semicolon (;).
'value1;value2;value5'
When updating a record through the Contacts API, the property value should be a single string with all of the values separated by semicolons:
{
"property": "property_name",
"value": "value1;value3;value4"
},
For the Forms API, make sure that the semicolons are also URL encoded:
...&property_name=value1%3Bvalue2%3Bvalue4