Auto-complete Contact Reference fields in a Contribution form profile are not saving. Hidden field values forgotten after confirmation page, triggering [Select a valid contact] message

The problem CiviContribute form with the custom data contact reference fields on it is at: http://208.88.5.242/cms/civicrm/contribute/transact?reset=1&id=3

I can fill in the form and use the control (Workplace 1, which is custom_60 field; or Workplace 2, which is custom_65 field) to select a name/value. Press "Continue >>".

On confirmation page hidden fields have correct values. (Other required fields may trigger errors at this stage as is normal).

<input name="qfKey" value="b73f696d65ec15cd6ee8f4fc4fa0301e_3962" type="hidden">
<input name="contact_sub_type_hidden" value="Member" type="hidden">
<input id="custom_60_id" name="custom_60_id" value="14" type="hidden">
<input id="custom_65_id" name="custom_65_id" value="16" type="hidden">
<input name="_qf_default" value="Main:upload" type="hidden">
<input name="MAX_FILE_SIZE" value="10485760" type="hidden">

After pressing "Confirm Contribution" button I get error page, and, I see the hidden variables are empty:

<input name="qfKey" value="b73f696d65ec15cd6ee8f4fc4fa0301e_7586" type="hidden">
<input name="contact_sub_type_hidden" value="Member" type="hidden">
<input id="custom_60_id" name="custom_60_id" value="" type="hidden">
<input id="custom_65_id" name="custom_65_id" value="" type="hidden">
<input name="_qf_default" value="Confirm:next" type="hidden">

Since form has values for <input id="cusotm_60" ...> etc, I presume this triggers error.

Interestingly, forgetting a required field causes only that error to be reported on confirmation page, suggesting that validation of custom_60 & custom_65 fields was performed successfully on that first form page. Its only on confirmation submit that this "Select a valid contact" error pops up.

Also, as an admin I can edit a contact recordset using this contact reference field without a problem - selected values are saved to the record.

I've done one custom code modification - simply to enable the /civicrm/ajax/rest? ... url for the auto-complete call to work for anonymous users.

Anyone's attention to this is much appreciated!