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Apparently QuantLib uses the start date of the inflation swap only for the purpose to calculate the relevant base date for the referenced inflation index.
As a result, a non-zero price may be produced for a swap maturing before its own start date.

Click on
download to download an xml file that contains an Excel formula that demonstrates this issue.

You may then reproduce the issue in spreadsheet by clicking on Go -> Load Excel Formula from Text File or one of its variations and choose the xml file downloaded in the step above.